We follow Bickel [1]} and Cai et al. [2]} and consider estimating the covariance matrix \({\mathbf {\Sigma }}\)  over the parameter space \(\mathcal {U}_\alpha = \mathcal {U}_\alpha (M_0, M) \)  where
\(\mathcal {U}_\alpha (M_0, M) = \left\lbrace {\mathbf {\Sigma }}: \max _j \sum _i\lbrace |\sigma _{ij}|: |i-j| > k\rbrace  \le Mk^{-\alpha } \text{ for all } k, \Vert {\mathbf {\Sigma }}\Vert \le M_0 \right\rbrace .\) 
